2007-05-03 IEEE C802.16m-07/097

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Project IEEE 802.16 Broadband Wireless Access Working Group <http://ieee802.org/16>

Title Link Performance Abstraction based on Mean Mutual Information per Bit (MMIB) of the LLR Channel

Date Submitted 2007-05-02

Source(s)

Krishna Sayana, Jeff Zhuang,

Ken Stewart

Motorola Inc 600, N US Hwy 45, Libertyville, IL

Re: Call for comments on Draft IEEE802.16m Evaluation Methodology Document

Abstract Link abstraction based on MMIB without the need of adjustment factors and extendable to MIMO ML receivers

Purpose For discussion and approval by TGm

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Contents 1

2 3 1.0 Purpose................................................................................................................................................. 3 4 2.0 Introduction.......................................................................................................................................... 3 5 3.0 Link Performance Prediction using MMIB – Overview...................................................................... 4 6 4.0 Concept of Bit LLR Channel ............................................................................................................... 6 7 5.0 MIB Mapping for Single Input Single Output Systems (SISO) .......................................................... 6 8

5.1 Mutual Information Computation – BPSK/QPSK............................................................................ 8 9 5.2 Mutual Information Computation – M-QAM................................................................................... 9 10

6.0 Generalized LLR PDF Model - Mixture of Gaussians ...................................................................... 10 11 6.1 Numerical Simulation to Obtain LLR PDFs and MIBs.................................................................. 12 12

7.0 MMIB Link Abstraction for SISO/SIMO – Detailed Description of the Simulation Step................ 12 13 8.0 BLER Mapping Function – Detailed Description and Numerical Approximations .......................... 13 14 9.0 Performance Prediction for HARQ.................................................................................................... 16 15

9.1 Chase Combining ................................................................................................................... 16 16 9.2 Incremental Redundancy ................................................................................................................ 16 17 9.3 Numerical Results with SISO ......................................................................................................... 17 18

10.0 MIMO Mapping based on SISO MMIB Mapping Functions........................................................... 18 19 10.1 Linear Receivers ................................................................................................................... 18 20 10.2 Successive Cancellation for Non-Linear Receivers..................................................................... 18 21 10.3 Eigen Decomposition with Channel Knowledge for Non-Linear Receivers............................... 19 22

11.0 Non-Linear Receiver Modelling ...................................................................................................... 20 23 12.0 Conclusions...................................................................................................................................... 22 24 References................................................................................................................................................. 22 25

26

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Link Performance Abstraction based on Mean Mutual Information 1 per Bit (MMIB) of the LLR Channel 2

Krishna Sayana, Jeff Zhuang, Ken Stewart 3 Motorola Inc 4

1.0 Purpose 5

This contribution provides detailed description of a link abstraction technique. The level of details herein 6 was absent in the Draft IEEE 802.16m Evaluation Methodology Document, even though the concept 7 was alluded on page 40 (line 19) to 42 (line 6). Hence, the details are provided and explained to allow 8 simulation study to be conducted to verify/improve the proposed method. 9 10 In summary, with the proposed modeling technique, accurate link abstraction can be obtained based on a 11 mean mutual information per coded bit (MMIB) metric which is the mean mutual information between 12 coded bits and their LLR values. The known Mutual Information/Capacity ESM method for link 13 abstraction has a closed-form expression for BPSK/QPSK. But for 16QAM/64QAM, some empirical 14 compensation factor must be introduced, just like in the well-known EESM method. On the other hand, 15 the MMIB metric itself, once computed for QPSK, 16QAM, and 64QAM, can be used to model the 16 decoded performance for any MCS and coding rate, without the need of defining any MCS-dependent 17 adjustment factors. The method is also extended to model HARQ (both chase and IR) and MIMO ML or 18 quasi-ML receiver. 19

2.0 Introduction 20

Methods of block error rate (BLER) prediction conditioned on measurable physical parameters such as 21 signal-noise ratio and multi-path channel state are required for modeling a link performance in system 22 level simulation. A well-known approach to link performance prediction is the Effective Exponential 23 SINR Metric (EESM) method. This approach has been widely applied to OFDM link layers [2][3][6], 24 but this approach is only one of many possible methods of computing an ‘effective SINR’ metric. 25 26 One of the disadvantages of the EESM approach is that a normalization parameter (usually represented 27 by a scalar, β ) must be computed for each modulation and coding (MCS) scheme. In particular, for 28 broader link-system mapping applications, it can be inconvenient to use EESM when combining 29 codewords mapped onto different modulation types, where the EESM method can require the use of so-30 called symbol de-mapping penalties. Seeking a means to overcome some of the shortcomings of EESM, 31 we focus here on the Mutual Information based approach to link performance prediction. 32 33 The first part of the contribution focuses on the single-stream transmissions where a single equivalent 34 channel can be readily constructed (e.g.., Single Input Single Output (SISO), MISO with 802.16e Matrix 35 A encoding, or simply a SIMO channel). In this case, the proposed approach links the SINR of each 36 subcarrier (or group of sub-carriers) to the mutual information between each encoded bit comprising the 37 received QAM symbols and the corresponding log-likelihood ratio (LLR). This yields a mean mutual 38 information per bit (MMIB) measure that may be attributed to “quality” of the entire codeword if sent 39 on that particular channel. This measure can then be used to predict the block error rate for a 40 hypothesized MCS transmission, or space-time coding scheme, or spatial multiplexing scheme. In this 41

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document, we derive an appropriate mutual information measure for each component bit comprising the 1 QAM symbol by using Monte Carlo integration and then show the adjustment parameter β for each 2 MCS can be avoided. 3 4 In the second part of this contribution, we extend the MMIB method to open loop MIMO links (e.g., 5 802.16e Matrix B mode). Link performance prediction methods such as EESM may be extended to 2x2 6 MIMO systems, by reducing the MIMO channel to two equivalent SISO channels and associating an 7 SNR metric to each such channel. However, with a quasi maximum likelihood (QML) receiver, such as 8 a receiver constructed from the general class of sphere decoding methods, this type of separation is not 9 justified. Approximations can be made assuming a) a perfect successive cancellation receiver or b) 10 eigenmode transmission (i.e. assuming a known channel) at the transmitter, but we will show they 11 cannot accurately model the true QML/ML receiver performance. MMIB-based approach parameterized 12 by three variables is shown to have very good prediction. 13

3.0 Link Performance Prediction using MMIB – Overview 14

For communication systems like OFDM where multiple channel states may be obtained on a transmitted 15 codeword, link performance prediction, in general, is based on determining a function 1 2( , , )I SINR SINR 16 which maps multiple physical SINR observations (or more generally of the channel states itself for 17 MIMO channels as we will show later) into a single “effective SINR” metric effSINR (or equivalent) 18 which can then be input to a second mapping function ( )effB SINR to generate a block error rate (BLER) 19 estimate for a hypothesized codeword transmission. We assume the access to a set Ω of N SINR 20 measures, denoted nSINR , 0 n N≤ < . Note that the precise definition of these observations will depend on 21 the SISO/MIMO transmission mode and a receive type, but for the simple SIOS case, the SINR 22 measures may be assumed to correspond to SINR observations of individual data sub-carriers (and 23 therefore of associated QAM symbols) transporting the hypothesized codeword of interest. 24 25 The first mapping function I , and effective SINR metric effSINR , may be generally defined as 26

11 2

1 Neff n

n

SINR SINRI INα α=

⎛ ⎞ ⎛ ⎞Γ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∑ (0.1) 27

where 1α and 2α are constants (and maybe constrained to be equal), which may be MCS-specific, and Γ 28 may correspond to a defined statistical measure. ( ).I is a reference function usually selected to represent 29 a performance model. Exponential ESM is derived by using an exponential function, which is based on 30 using Chernoff approximation to the union bounds on the code performance. Similarly other 31 performance measures like capacity or mutual information can be used. The accuracy of the model to 32 some extent is dependent on how closely the reference model represents the code performance (with 33 sufficient parameterization a given model can yield a reasonably good accuracy as in EESM). 34 35 In the method proposed here, Γ is the mean mutual information per coded bit (MMIB), or simply 36 denoted as M , and 1α and 2α are discarded (i.e. set to unity). That is, equation (0.1) becomes 37

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( ) ( )

( )

1

1 1

1

1

1( )

N

eff m nn

N

eff m nn

M I SINR I SINRN

SINR I M I I SINRN

=

− −

=

= =

⎛ ⎞⇒ = = ⎜ ⎟⎝ ⎠

∑

∑ (0.2) 1

where (.)mI is a function that depends on the modulation type identified by m and the associated bit 2 labeling in the constellation1, where 2,4,6m∈ corresponding to QPSK, 16-QAM, 64-QAM respectively. 3

(.)mI maps the sub-carrier SINR to the mean mutual information between the log-likelihood ratio and the 4 binary codeword bits comprising the QAM symbol. 5 6 Due to the asymmetry of bit-to-symbol mapping in the constellation, each bit in the m -tuple labeling of 7 each QAM symbol perceives a different equivalent channel (commonly referred to as unequal error 8 protection). An equivalent bit channel is defined and appropriate bit-wise measures are derived in the 9 following sections. In fact, the average of the mutual information of these bit-wise mutual information 10 measures is derived. More precisely, for an m -tuple input word there exist m mutual information 11 functions2 ,m iI , where 12

,1

1( ) ( )m

m m ii

M I SINR I SINRm =

= = ∑ (0.3) 13

Note that mI may be approximated using numerical methods, and then stored for later use in link 14 performance prediction. We will refer to the above quantity as Mutual Information per coded Bit or MIB, 15 with the understanding that it is derived by averaging over the “m” bit channels. Furthermore, mean 16 mutual information per bit (MMIB) is used to refer to the mean obtained over different channel states or 17 SNR measures. We will show how mI can be accurately computed without the need of defining any 18 adjustment factor and only three mI functions (i.e., for m=2,4,6) need to be specified that do not depend 19 on the coding rate. The three MMIBs are adequate for predicting performance for any modulation and 20 coding scheme. 21 22 The second functional relationship necessary to estimate the BLER – that is, the BLER function ( )B M – 23 may be derived simply from the performance of a specific coding type and decoder under AWGN 24 conditions. Typically, a distinct function ( )B M is required for each possible MCS type supported for 25 system simulation. In later sections, means of generating and storing ( )B M , and simplifications to reduce 26 the number of distinct functions required for storage are discussed. 27 28 The basic SISO MMIB method described above can be readily extended to SIMO (i.e. 1x2, or MS 29 diversity) and MISO (the 802.16e Matrix A space-time code) channels by the application of the 30 appropriate MS combining operations. These modes result in a single SINR value per QAM symbol and 31 equation (0.2) can then be simply re-applied. 32 33 The extension of MMIB method in the case of linear MIMO receiver is straightforward. However, an 34 SINR measure cannot be obtained per QAM symbol in the case of Maximum Likelihood (ML) or 35 Sphere Decoding receivers. This presents a challenging problem, and we will show in later sections that 36 1 The constellation mappings in this document are specified in Section 8.4.9.4 ,[1]. 2 Only / 2m of these measures are distinct, due to the quadrature symmetry of 802.16e constellations.

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with the definitions and models we introduce here, a very accurate abstraction can be obtained for these 1 receivers without reducing the MIMO channel to two parallel SISO channels. This is another advantage 2 of MMIB-based link abstraction. 3

4.0 Concept of Bit LLR Channel 4

In general, the accuracy of a mutual information based metric depends to a large extent on the equivalent 5 channel over which this metric is defined. For example, a modulation constrained capacity metric is the 6 mutual information of a “symbol channel” (i.e., constrained by the symbol constellation). It is possible 7 to obtain a mutual information per bit metric from the symbol channel by simply normalizing this 8 constrained capacity (i.e, by dividing by the modulation order [1]). 9 10 However, given that our goal is to abstract the performance of the underlying binary code, the closest 11 approximation to the actual performance is obtained by defining an information channel at the coder-12 decoder level, i.e., defining the mutual information between bit input (into the QAM mapping) and LLR 13 output (out of the LLR computing engine at the reciever), as shown below. The concept of bit channel 14 encompasses MIMO channel and receiver. We will demonstrate that this definition will greatly simplify 15 PHY abstraction by moving away from an empirically adjusted model and introducing instead MIB 16 functions of equivalent bit channels. 17 18

19 20 In the bit channel above, the task now is to define efficient functions that capture the mutual information 21 per bit. The following sections further develop an efficient approach for MIB computation by 22 approximating the LLR PDF with a mixture Gaussian PDFs. We will begin with the development of 23 explicit functions for MIBs in SISO and later extend to MIMO. 24 25

5.0 MIB Mapping for Single Input Single Output Systems (SISO) 26

This section describes MIB defintion for SISO systems, focusing on the theoretical concepts and 27 notations. The numerical expressions/approximations for the actual MIB mapping functions for 28 implementation purposes are elaborated in the next section. 29 30 After the encoding step using the CTC or CC encoders to generate a binary codeword bit stream kc , the 31 QAM modulation step can be represented as a labeling map : Xµ Α→ , where Α is the set of m -tuples – 32

2,4,6m∈ – of binary bits and X is the constellation. Given the observation ny corresponding to the thn 33 QAM symbol, the demodulator computes the log-likelihood ratio (LLR) ,( )i nLLR b of the thi bit 34 comprising the symbol via the following expression (where the symbol index n is dropped for 35 convenience) 36

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( 1| ) ( | 1)( ) ln ln( 0 | ) ( | 0)

i ii

i i

P b y P y bLLR bP b y P y b

⎛ ⎞ ⎛ ⎞= == =⎜ ⎟ ⎜ ⎟= =⎝ ⎠ ⎝ ⎠

(0.4) 1

The computed LLR’s may then be input to a BCJR (or similar) decoder. When the coded block sizes are 2 very large in a bit-interleaved coded system (BICM), the bit interleaver effectively breaks up the 3 memory of the modulator, and the system can be represented as a set of parallel independent bit-4 channels [4]. Conceptually, the entire encoding process can be represented as follows: 5 6

7 Figure 1 – Conceptual .16e encoding & decoding process with a BICM model. 8

9

Due to the asymmetry of the modulation map, each bit location in the modulated symbol experiences a different ‘equivalent’ 10 bit-channel. In the model, each coded bit is randomly mapped (with probability 1/ m ) to one of the m bit-channels. The 11 mutual information of the equivalent channel can be expressed as: 12

1

1( , ) ( , ( ))m

i ii

I b LLR I b LLR bm =

= ∑ (0.5) 13

where ( , ( ))i iI b LLR b is the mutual information between input bit to the QAM mapper and output LLR for 14 thi bit in the modulation map. 15

16 More generally, however, the mean mutual information – computed by considering the symbol 17 observations at all N sub-carriers over the codeword – may be computed as 18

1 1

1 ( , ( ))N m

i in i

M I b LLR bmN = =

= ∑∑ (0.6) 19

The mutual information function ( , ( ))i iI b LLR b is, of course, a function of the QAM symbol SINR, and so 20 the mean mutual information M may be alternatively written 21

( ),1 1 1

1 1( )i

N m N

m b n m nn i n

M I SINR I SINRmN N= = =

= ∑∑ ∑ (0.7) 22

The mutual information function is in turn dependent on the SINR (itself a function of the sub-carrier 23 index n ) and the code bit index i , and varies with the constellation order m . Accordingly, the 24

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relationship , ( )im bI SINR is required for each modulation type and component bit index in order to 1

construct ( )mI SINR .3 2

5.1 Mutual Information Computation – BPSK/QPSK 3 4 Generally, if ( )H X is the entropy of X , then 5 6

( , ) ( ) ( | )I b LLR H b H b LLR= − (0.8) 7 That is, the mutual information between the coded bit value b and the LLR is equal to the uncertainty 8 concerning b (which is assumed to be unity) minus the uncertainty concerning b given that LLR is 9 available. But, clearly ( ) 1H b = and so 10

2

2

1( | ) ( | 1) log [ ( | 1)]2

1 ( | 0) log [ ( | 0)]2

LLR LLR

LLR LLR

H LLR b p z b p z b dz

p z b p z b dz

∞

−∞

∞

−∞

= = =

+ = =

∫

∫ (0.9) 11

12 However, the required mutual information function can also be expressed in a more convenient form for 13 numerical evaluation, specifically 14

20,1

1 2 ( | )( , ) ( | ) log 2 ( | 0) ( | 1)

LLRLLR

LLR LLRb

p z bI b LLR p z b dzp z b p z b

+∞

= −∞

⎛ ⎞= ⎜ ⎟= + =⎝ ⎠∑ ∫ (0.10) 15

where z is a dummy variable equal to LLR. 16 17 The received signal can then be represented as 18

y x n= + (0.11) 19 where 2[| | ] 1E x = - bit ‘0’ is transmitted as ‘+1’ and ‘1’ as ‘-1’- and 2 2[| | ] 1/(2 / )n s oE n E Nσ= = , / 2oN being 20 the noise variance per complex dimension. Substituting in equation(0.11), it can be easily shown that 21 the LLR simplifies to 22

( )22

nLLR x n

σ= + (0.12) 23

i.e., it is a scaled value of the received signal and is thus Gaussian, conditioned on a specific value of x, 24 where 22 / nµ σ= (conditioned on 1x = ) and 2 24 / 8 /n s oE Nσ σ= = are the mean and the variance of the LLR 25 respectively. Since the LLR satisfies 2 / 2µ σ= , the above expression simplifies to 26

( ) ( )

2 2

2( / 2)

22

1( , ) 1 log (1 )2

( ) 8 / 8

zz

s o

I b LLR e e dz

J J E N J SINR

σσ

πσ

σ

+∞ −−

−

−∞

= − +

= = =

∫ (0.13) 27

The above expression4 can be computed numerically (see [5]), and appears in Figure 2. 28

3 Note that in the 802.16e specification, bit indexing typically proceeds from 0. 4 Note that (.)J is not related to the well-known Bessel function of the first kind conventionally designated (.)nJ , where n denotes the function order.

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1 Figure 2 – MIB vs. Es/No (dB), BPSK/QPSK5 2

-20 -10 0 10 20 30 400

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

LLR

PD

F(LL

R |

b=1)

b0

-10 -5 0 5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

0.12

LLR

PD

F(LL

R |

b=1)

b1, b0

3 Figure 3 – BPSK and QPSK bit-wise conditional LLR distributions 4

5 We can note that, as expected, for BPSK, the LLR distribution is Gaussian with mean 6

22 / 4 / 12.65n s oE Nσ = = (SNR = 5 dB). Predictably for QPSK, the distribution is also Gaussian with a 7 mean which is one half of the BPSK mean. 8

5.2 Mutual Information Computation – M-QAM 9 BPSK/QPSK MIB is obtained by a known closed-form expression. It is clear that a corresponding non-10 linear function exists for higher order QAM. Before proceeding with determining these with the 11 proposed LLR channel model in the next section, we briefly discuss possible approaches that can be 12 considered to obtain similar functions. 13

• ESM with BPSK MIB (MIESM): Simply use the BPSK MIB function in place of exponential in 14 EESM. This approach would require “beta” parameterization adjustment similar to EESM for 15 higher order modulation. 16

5 This does not mean it is the same function for both BPSK, QPSK. The difference is only a fixed scaling of SNR by 3 dB.

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• Constrained Capacity with a Modulation (Ungerboeck). Also referred to as CM (Coded 1 Modulation Capacity). This model does not accurately reflect performance in a fading channel 2 with coding and interleaving. Further, it does not take into account the constellation mapping. 3

• Bit-Interleaved Coded Modulation (BICM) Capacity (Caire). This model captures the capacity of 4 each bit channel in an interleaved coded modulation, and reduces to the general LLR channel 5 model developed here for SISO. Non-linear functions must be considered for bit channel 6 capacities/MI with a given modulation constellation 7

• Proposed Approach (see detail in the next section): With the LLR model, the framework is 8 general and it is applicable to all cases including MIMO since it uses the baseline receiver 9 models to approximate the MIB with actual decoding. Numerical characterization of functions is 10 required, but such characterization is based on a simple transmission and the receiver models and 11 does not require any exhaustive link simulations (the end result is verified). 12

13

6.0 Generalized LLR PDF Model - Mixture of Gaussians 14

It is shown that there exists a known closed form expression for BPSK. We now derive functions of 15 similar complexity for higher order modulations. The LLR PDF of 16QAM is shown in the figure below 16 for SNR = 5 dB for all the four (m=4) component bits. It is shown in the figure, that it can be 17 approximated as a mixture Gaussian distribution with two component Gaussian distributions defined by 18 individual means, standard deviations and the associated marginal probability. If similar PDF is plotted 19 for higher SNRs, it can be seen that the component distributions do not overlap. Conceptually, it can be 20 easily proved using minimum distance arguments, that at asymptotically high SNRs, we will have a 21 mixture distribution composed of Gaussians in the conditional LLR PDFs (we will skip the proof for 22 brevity here). 23 24 The implications of this observation are profound, and it indicates that some kind of structure exists in 25 the non-linear MIB functions. In other words, if the LLR distribution can be approximated by a mixture 26 of Gaussian distributions (which are non overlapping), then it follows that the corresponding MIB can 27 be expressed as a sum of J(.) functions, which corresponds to the MIB for a Gaussian conditional LLR 28 PDF distribution. 29 30 31

1 iMixture of Gaussians ( ) ( ) 1

K

i i ii

I x a J c x a=

→ = =∑ ∑ 32

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-4 -2 0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

LLR

P(L

LR |

b=1)

Conditional DistributionGaussian distribution

Statistics ofdistributionmean = 1.26std = 1.57

1

-10 -5 0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

LLR

PD

F(LL

R|b

=1)

Conditional LLR distributionGaussian Distribution 1Gaussian Distribution 2Sum of two Gaussians

Statistics of twodistributions

mean = [1.45, 6.039]Std = [1.62, 2.69 ]prob = [0.59, 0.41]

2 Figure 4 - 16QAM : Conditional LLR distributions modeled as a mixture of 3

Gaussian distributions at SNR = 5dB a) b2,b0 b) b3,b1 4 5 The MIB is defined by considering a conditional hypothesis on all the individual bits. In the above 6 figure, LLR PDFs corresponding to two of the bits is a Gaussian distribution. The PDFs for other two 7 can be expressed as mixture of two Gaussian distributions. It is clear that that the MIB of 16QAM can 8 be represented by a sum of three J(.) functions. Similarly, other modulations orders can be expressed as 9 a mixture of Gaussian distributions and as the modulation order is increased could typically be 10 composed of greater than 3 Gaussians. 11 12 However, limiting the maximum to 3 dominant Gaussians is found to yield very good approximation to 13 the actual non-linear MIB function for typical cases of interest. The approximations are obtained by 14

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numerical optimization and are summarized in Table 1. The accuracy is verified in Figure 5 and the 1 deviation from the actual curve is less than 5e-3. 2 3 4 5

MI Function Numerical Approximation 2 ( )I γ (QPSK) (2 )M J γ= (Exact)

4 ( )I γ (16 QAM) 1 1 1(0.8 ) (2.17 ) (0.965 )2 4 4

M J J Jγ γ γ= + +

6 ( )I γ (64 QAM) 1 1 1(1.47 ) (0.529 ) (0.366 )3 3 3

M J J Jγ γ γ= + +

Table 1 – Numerical approximations for MMIB mappings. 6

-30 -20 -10 0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR(dB)

I(b,

LLR

)

16 QAM

Result from SimulationNumerical Approximation

0 5 10 15 20 25 30 35 400.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR(dB)

I(b,

LLR

)

64 QAM

Result from SimulationNumerical Approximation

7 Figure 5 - Comparison of Numerical Approximations with simulated results for 8

a) 16 QAM b) 64 QAM. 9

6.1 Numerical Simulation to Obtain LLR PDFs and MIBs 10 For reference, the following steps can be used for obtaining the above approximations 11 12 Step 1 LLR conditional PDF’s of all the bits for each specific modulation are obtained by 13 numerical simulation at each SNR for a scalar channel 14 Step 2 MIB is then obtained by numerical integration 15 Step 3 Approximation using sum of basis functions J(.) by curve fitting considering all SNRs 16

7.0 MMIB Link Abstraction for SISO/SIMO – Detailed Description of 17 the Simulation Step 18

The current 802.16e ECINR reporting requirement (802.16e Table 298a, Section 8.4.5.4.10.4) lists the 19 MCS levels specified in Table 2 (to be updated based on simulation methodology support for different 20 MCSs, packet sizes etc., specific to 16m). 21 22

MCS Label Modulation Code

Rate Repetition

Factor

Max. Inf. Word

Length

Max. Code Word Length

(bits)

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(bits) 1c QPSK 1/2 6 1b QPSK 1/2 4 1a QPSK 1/2 2

See MCS 1

1 QPSK 1/2 1 480 960 2 QPSK 3/4 1 432 576 3 16-QAM 1/2 1 480 960 4 16-QAM 3/4 1 432 576 5 64-QAM 1/2 1 432 864 6 64-QAM 2/3 1 384 576 7 64-QAM 3/4 1 432 576 8 64-QAM 5/6 1 480 576

Table 2 –Example MCS Set for Simulation. 1 2 The associated codeword lengths are adopted to be the maximum codeword lengths possible 3 corresponding to each modulation and code rate combination (for illustrative purposes). Accordingly, 4 for the purpose of MMIB based link abstraction, we require 5 6

1) Mutual information mapping functions ( )mI x for all modulation types QPSK, 16-QAM, 64-7 QAM to obtain MMIB from a given channel realization, and 8

2) A block error rate (BLER) mapping function ( )B Mϕ for mapping MMIB to a predicted BLER, 9 where ϕ is the index identifying the codeword length and code rate. 10

11

The second requirement is based on the observation that MMIB to block error rate mapping is found to 12 be independent of the modulation itself to a good approximation (see below). 13

8.0 BLER Mapping Function – Detailed Description and Numerical 14 Approximations 15

18 The BS can store the AWGN reference curves for different MCS levels in order to map the MMIB to 19 BLER. Another alternative is to approximate the reference curve with a parametric function. For 20 example, we consider a Gaussian cumulative model with 3 parameters which provides a close fit to the 21 AWGN performance curve, parameterized as 22

1 , 02 2a x by erf c

c⎡ ⎤−⎛ ⎞

= − ≠⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

(0.14) 23

where a is the “transition height” of the error rate curve, b is the “transition center” and c is related to 24 the “transition width” (transition width = 1.349 c ) of the Gaussian cumulative distribution. In the linear 25 BLER domain, the parameter a can be set to 1, and the mapping requires only two parameters, which 26 are given for each MCS index in the table below. 27 28

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The accuracy of the curve fit with this model is verified in below with MCS modes in 802.16e. 1 2

Modulation Code Rate

b c

QPSK 1/2 0.5512 0.0307 16QAM 3/4 0.7863 0.03375 64QAM 5/6 0.8565 0.02622

Table 3 - Parameters for Gaussian cumulative approximation 3 to BLER mapping. (Block Size = 480 bits) 4

0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0 0.1

0.2 0.3

0.4 0.5

0.6

0.7 0.8

0.9 1

MMIB

BLER MCS 1 SimulationMCS 1 Gauss Cum ApproxMCS 2 SimulationMCS 2 Gauss Cum ApproxMCS 3 SimulationMCS 3 Gauss Cum Approx

5 Figure 6 - Curve fit for BLER mapping. 6

7 So for each MCS the BLER is obtained as 8

1 1 , 02 2

MCSMCS

MCS

x bBLER erf c

c

⎡ ⎤⎛ ⎞−= − ≠⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(0.15) 9

10 Further, we can achieve an additional simplification. The following figure plots MMIB vs BLER (i.e. 11

( )B Mϕ ) for numerical results obtained in 802.16e simulations using 6 different MCS’s with rates 1/2 and 12 3/4 on an AWGN channel. 13

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0 0 .1 0.2 0.3 0 .4 0 .5 0.6 0 .7 0 .8 0 .9 1-25

-20

-15

-10

-5

0

M M IB

BLE

R

Q P S K R = 1/2

16Q A M R = 1/2

64Q A M R = 1/2

Q P S K R = 3/4

16Q A M R = 3/4

64Q A M R = 3/4

1 Figure 7 - BLER mappings for MMIB from AWGN performance results. 2

3 It can be seen from the figure that – to a first-order approximation – the mapping from MMIB to BLER 4 can be assumed independent of the QAM modulation type. However, since code performance is strongly 5 dependent on code sizes and code rates, ( )B Mϕ will not be independent of these parameters. 6 7 With the above result, we generalize the AWGN reference curves to be a function of the block size 8 and coding rate (BCR) 9

1 1 , 02 2

BCRBCR

BCR

x bBLER erf c

c

⎡ ⎤⎛ ⎞−= − ≠⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(0.16) 10

With this simplification, a base station needs to store two parameters for each supported BCR mode. 11 12 Note: The choice of this particular MMIB to BLER mapping is due to the underlying physical 13 interpretation. The parameter b is closely related to the binary code rate and will be equal to the code 14 rate for an ideally designed code. Similarly, parameter c represented the rate of fall of the curve and is 15 also related to the block size. 16 Note 2: . It is also possible to express these parameters as simple 2-dimensional parameterized functions 17 of block size and code rate as follows, which could further reduce storage requirements and streamline 18 simulation methodology. 19

'( , ) ( , )( , )

b f R L R f R Lc g R L= = +=

20

21

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9.0 Performance Prediction for HARQ 1

It is clear that once compted, MMIB metric relates to the underlying coder-decoder performance and is 2 independent of the modulation order transmission modes etc., Due to this property, link prediction for 3 HARQ can be accomplished easily when the multiple (re)transmissions corresponding to an information 4 packet do not correspond to the same modulation. This is illustrated in the above section, where we have 5 shown that the MMIB to BLER mappings are independent of the modulation order – to a good 6 approximation. 7

8 This allows the transmitter to predict performance with hybrid ARQ, even when the retransmissions 9 support modulation and transmission modes different from the first transmission. We outline the general 10 approach here for performance prediction with HARQ 11

9.1 Chase Combining 12 This is a straight-forward extension since the post-processing SNRs can be obtained as simple sum of 13 the SNRs on the first transmission and subsequent retransmissions. 14

1 1

( )qN

m iji j

MMIB I γ= =

=∑ ∑ 15

where jγ is the i th symbol SNR during j th retransmission. 16

9.2 Incremental Redundancy 17 With IR, typically a transmitter transmits packets which are components of a mother code. Given a 18 packer is received in error, the BS tries to transmit independent code information as much as possible to 19 maximize coding gains at the receiver. Typically, only when these combinations are exhausted, a 20 retransmission of previous packet transmissions is performed. In this context, the performance of the 21 decoder at each stage is that corresponding to a binary code with the modified equivalent code rate and 22 code size (as shown below), and neglecting any partial repetition of previously sent packets for 23 modelling purposes. 24 25

First TransmissionMMIB1

Second RetransmissionMMIB2

C1 Code Bits C2 Code Bits

X Information Bits

1 2

1 2

1 1 2 2,2

1 2

Effective Code Rate =

Effective Code Size =

HARQ

xc cc c

c MMIB c MMIBMMIBc c

+++

=+

Inputs to BLER Mapping Block for Lookup and Prediction

26 Figure 8 – MMIB Update after a Retransmission and the Required Parameters for BLER Lookup 27

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1

The performance prediction can be performed by combining the MMIBs on the transmissions as shown 2 in the above figure, and looking up the BLER relationship corresponding to the modified effective code 3 rate and code size. 4

Note: A code rate-code size parameterized relationship for b,c parameters in the AWGN reference (see 5 note at the end of previous section), is clearly very helpful to cover the new possible combinations with 6 IR. 7

[To be inserted in future. The supported IR modes and the corresponding parameters in 16m modeling.] 8

[To be inserted by another contribution: The partial IR modeling] 9

9.3 Numerical Results with SISO 10 The following results show the performance prediction accuracy of EESM and MMIB approaches. 11 Optimal beta is used for EESM obtained by link simulations. The proposed ‘sum of Gaussians’ 12 mappings are used for MMIB and no further fudge factors are used. It is clear that performance 13 prediction is close to (slightly better) EESM. Further, MMIB approach is more robust to channel models 14 etc., variation compared to EESM. 15

Note that the “Effective SNR” in the plot is the SNR of the reference (AWGN) curve which results in 16 the same FER as the given fading channel realization. The curves are plotted in an effective SNR 17 domain for comparison purposes only. For MMIB mapping, we can operate in MMIB domain directly. 18

19

14 14.5 15 15.5 16 16.5 17 17.5 18 18.5 19-1.5

-1

-0.5

0 64QAM R=5/6 Matrix A STC - EESM With Beta Optimization

20

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14 14.5 15 15.5 16 16.5 17 17.5 18 18.5 19-1.5

-1

-0.5

0

Effective SNR

BLE

R

64QAM R=5/6 Matrix A STC, MMIB - No Beta Parameters

TU Channel Realizations

AWGN Reference

1 Figure 9 – Performance Prediction for SISO with a) EESM b) MMIB 2

10.0 MIMO Mapping based on SISO MMIB Mapping Functions 3 We briefly discuss the approaches that may be used to derive the mappings for Matrix B mode. Note 4 that the mapping for matrix A is fairly straightforward once the post processing SNR with STC is 5 derived. 6

10.1 Linear Receivers 7 With linear receivers like MMSE, each MIMO channel is treated as two equivalent SISO channels with 8 SNRs given by post combining SNRs of the linear receiver. The MIB can be obtained as 9

1 1

1 ( )

( )

tNN

m ijt i j

M INN

BER B Mϕ

γ= =

=

=

∑∑ (0.17) 10

where ijγ is the post combining SNR of layer j on subcarrier i , tN is the number of transmit antennas, N 11 is the total number of coded subcarriers, and the mapping functions (.)mI and (.)Bϕ are defined in 12 sections on SISO for each MCS. 13 14

10.2 Successive Cancellation for Non-Linear Receivers 15 Successive cancellation approaches can be considered for decoding of SM schemes (e.g., matrix-B) and 16 give improved performance compared to linear receivers. ZF and MMSE based approaches can be 17 considered. Here, we summarize the algorithm with QR decomposition (equivalent to ZF approach). 18 19

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The QR decomposition of the channel matrix is given by 1

=H QR (0.18) 2

where Q is a 2x2 unitary matrix and R is a 2x2 upper triangular matrix. By pre-multiplying the received 3 vector with HQ , we obtain 4

H H= +Q y Rs Q ny' = Rs + n'

(0.19) 5

where 2[ ] [ ' ']H HE E Iσ= =n n n n , where 2σ is the variance per receive antenna and I is the 2x2 identity 6 matrix. With this transformation, the second symbol has no interference from the first symbol and the 7 LLRs can be computed from the following equation 8

2 22 2 2' 'y R s n= + (0.20) 9

A hard or soft estimate of the second symbol can be used to cancel the interference and decode the first 10 symbol. 11

1 11 1 12 2 12 2 2ˆ' 'y R s R s R s n= + − + (0.21) 12

Assuming perfect cancellation, the SNRs of the two layers are given by 13

2 2

11 221 22 2

| | | |, R Rγ γσ σ

= = (0.22) 14

and the MIB mapping can be obtained using the SISO MIB mappings as follows 15

2

1 1

1 ( )2

N

m iji j

M IN

γ= =

= ∑∑ (0.23) 16

The optimal ordering for cancellation requires maximizing the SNR of the first detected layer. This can 17 be done by permuting the columns of H , and choosing the best possible QR decomposition.6 18

10.3 Eigen Decomposition with Channel Knowledge for Non-Linear 19 Receivers 20 Define HW = H H (or HHH ). W is a 2x2 random non-negative matrix and has real non-negative eigen 21 values. The capacity can then be written in terms of eigen values 1 2,λ λ of W , 22

2

2

1

log det( * )

log(1 * )

H

ii

C I SNR

SNRλ=

= +

= +∑

H H (0.24) 23

The capacity of an instantaneous channel matrix remains the same, regardless of whether the data is 24 transmitted on eigenmodes or not, assuming that no water-filling is allowed on the eigenmodes. 25 However, with linear receivers, pre-coding/beamforming on eigen modes allows us to achieve channel 26 capacity without added implementation complexity of a non-linear receiver. 27 6 For higher number of receive and transmit antennas, reduced complexity ordering algorithms are proposed, but they are not required for a 2x2 system.

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With this observation, we now assume that data is transmitted along the eigen modes, and treat each of 1 the modes as a separate layer, which allows us to employ the MMIB models developed for SISO 2 channels. The MMIB mapping is given by 3

2

1 1

1 ( )2

N

m iji j

M IN

λ= =

= ∑∑ (0.25) 4

where (.)mI are the MMIB mappings for a SISO system. Numerical approximations for these functions 5 are provided as before. Note that this is an approximate model, since the arguments are based on 6 capacity, and does not exactly capture the performance of an ML receiver with non-Gaussian 7 constellations which are used in practice. 8

11.0 Non-Linear Receiver Modelling 9

The most accurate way to model the performance of non-linear receivers is to operate in the MIB 10 domain itself, without requiring an SNR interpretation. In other words, we will deal with the LLR 11 channel directly with the hypothesis of an ML receiver. In general, we can think of MIB as now being 12 defined for a hyper-constellation induced by the instantaneous channel matrix. 13 14 In addition by imposing the structure of mixture Gaussian distributions, we identify three dominant 15 Gaussians corresponding to a channel matrix 16

1 2 3

3

1 2 31

[ , , ]

( ) ( ), 1i i ii

I c a a a a

γ γ γ

γ=

→

= + + =∑

H

H 17

18 Step 1: Determine the dominant Gaussian means by simple algebraic mappings from the channel matrix 19 Step 2: Determine the parameterized sum of Gaussians with these means by numerical simulation and 20 curve fit. 21 22 The effort required for Step 2 can be intensive, but once determined these functions are fixed and do not 23 have any runtime impact on simulation modelling. 24 25 The plots below compare EESM with Eigen decomposition and MMIB with the above approach 26 showing 15 different TU channel realizations. The spread of the blue curves represents the accuracy of 27 the performance prediction. 28

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9 10 11 12 13 -1.5

-1

-0.5

0 64QAM R=1/2 Matrix B 2x2

Effective SNR

FER

EESM with EigenDecomposition

Optimal β

TU ChannelRealizations

AWGN Reference

1

9 9.5 10 10.5 11 11.5 12 12.5 13 -1.5

-1

-0.5

0

Effective SNR

FER

64QAM R=1/2 Matrix B 2x2

TU ChannelRealizations

AWGN Reference

MMIB

2 Figure 10 – Performance prediction for a MIMO ML receiver with a) EESM with Eigen decomposition b) MMIB for 3

ML receivers 4 5

In this case, with EESM, the error in effective SNR evaluation is -1/+0.5 dB at 10% frame error rate. It 6 is -0.2/+0.1 dB with the MMIB mappings. It is further noted that similar result is obtained with EESM 7 when other mappings based on MMSE or SIC are used. It is clear using MMIB based mapping targeted 8 at non-linear receiver operation results in significant improvement compared to EESM. Using other 9 SISO based mappings, i.e. mutual information mapping itself would result in similar degradation. But 10

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the proposed approach is shown to have prediction accuracy similar to SISO, and with no additional beta 1 parameters specific to MCSs (the functions once defined for each modulation, are common for all 2 MCSs) 3

12.0 Conclusions 4

This contribution proposes an LLR based bit-channel model to define mutual information measures 5 applicable to both SISO and MIMO. Further, we have shown that MIB in all cases can be expressed as a 6 sum of Gaussian approximation, which allows us to implement MIB evaluation with simple functions 7 obtained and approximated numerically. MMIB approach permits accurate prediction of code 8 performance independent of modulation order and the channel (Validated with TU, PA, PB etc.) 9

Similar MMIB vs. BLER relationship is observed for TU channel and AWGN reference channels, which 10 avoids optimization of parameters required for EESM mapping. Further, a 2 parameter Gaussian 11 Cumulative curve fit is recommended for MMIB to BLER/FER mappings, due to its accuracy and 12 physical interpretation. 13

MMIB allows performance prediction when code words from different modulation orders are combined 14 for decoding in a HARQ systems. Additional parameters are not required for HARQ. Accurate 15 mappings are also developed for an ML receiver, which obtain MIB as a function of channel matrix 16 itself, without the need to generate parallel channel approximations. Further, the beta parameters of 17 these approximate models are typically sensitive to MIMO channel models used in link simulations. 18

In conclusion, MMIB is a highly accurate tool to study and compare system performance of advanced 19 MIMO receivers and transmission modes in 16m systems. 20

21

References 22

[1] IEEE Std 802.16 – 2004, “IEEE Standard for Metropolitan Area Networks - Part 16: Air Interface for Fixed Broadband 23 Wireless Systems” 24

[2] 3GPP TSG-RAN-1, Nortel Networks, "Effective SIR Computation for OFDM System-Level Simulations," Document 25 R1-03-1370, Meeting #35, Lisbon, Portugal, November 2003 26

[3] Ericsson, “System-level Evaluation of OFDM – Further Considerations”, R1-031303 27

[4] G. Caire, “ Bit-Interleaved Coded Modulation”, IEEE Transactions on Information Theory, Vol. 44, No.3, May 1998. 28

[5] J. Kim, A. Ashkhimin, A. Wijngaarden, E. Soljanin, “Reverse Link Hybrid ARQ: Link Error Prediction Methodology 29 Based on Convex Metric”, Lucent Technologies, 3GPP2, TSG-C WG3. 30

[6] Shawn Tsai, “Effective-SNR Mapping for Modeling Frame Error Rates in Multiple-State Channels”, Ericsson, 3GPP2-31 C30-20030429-010. 32

[7] K.Brueninghaus, D. Astely, Thomas Salzer, Samuli Visuri, “ Link Performance Models for System Level Simulations of 33 Broadband Radio Access Systems”. 34

[8] Lei Wan, Shiauhe Tsai, Magnus Almgren, “A Fading-insensitive Performance Metric for a Unified Link Quality 35 Model”, pg 2110-4, Proceedings of WCNC, 2006. 36